The Fundamental Theorem of Calculus: Derivatives and Integrals Are Opposites

Differentiation and integration are inverse operations.

This is the Fundamental Theorem of Calculus, and it's not an overstatement to call it the most important theorem in calculus. Maybe in all of mathematics.

Here's why it's shocking: derivatives and integrals seem like completely different ideas. Derivatives measure instantaneous rate of change — slopes of tangent lines, velocities at single instants. Integrals measure accumulated totals — areas under curves, distances traveled over time.

One is about the infinitely small. The other is about adding up infinitely many pieces.

Yet they're inverses. Undo one, you get the other. The theorem that proves this changed physics, engineering, economics, and every quantitative science.

The Two Parts

The Fundamental Theorem has two parts. Both are essential.

Part 1 (Derivatives Undo Integrals):

If you integrate a function and then differentiate the result, you get back the original function.

d/dx [∫ₐˣ f(t) dt] = f(x)

The derivative of an integral-with-variable-upper-bound equals the integrand evaluated at that bound.

Part 2 (Integrals Undo Derivatives):

If F(x) is an antiderivative of f(x) — meaning F'(x) = f(x) — then:

∫ₐᵇ f(x) dx = F(b) - F(a)

You can evaluate a definite integral by finding any antiderivative and subtracting its values at the bounds.

These two statements are really one insight from different angles: differentiation and integration are opposites.

Why Part 1 Is True: A Rate Intuition

Let A(x) = ∫ₐˣ f(t) dt — the area under f from a to x.

What happens when x increases by a tiny amount dx?

The area increases by a thin slice: approximately f(x) · dx.

So the rate of change of area is: dA/dx ≈ f(x) · dx / dx = f(x)

As dx → 0, this becomes exact: A'(x) = f(x).

The rate at which area accumulates equals the height of the function.

This makes physical sense. If f(t) represents velocity, then A(x) represents distance. The rate at which distance accumulates is... velocity. Obviously.

The theorem is really saying: accumulation and rate are opposite concepts, mathematically.

Why Part 2 Is True: The Shortcut

Part 2 is the practical miracle.

Computing integrals from the Riemann sum definition is brutal. You're taking limits of increasingly complex sums. Even for simple functions, it's tedious.

But Part 2 says: if you can find any function F whose derivative is f, then:

∫ₐᵇ f(x) dx = F(b) - F(a)

Instead of adding up infinitely many slices, you evaluate F at two points and subtract. That's it.

The proof follows from Part 1. If A(x) = ∫ₐˣ f(t) dt, then A'(x) = f(x). So A is an antiderivative of f.

If F is any other antiderivative of f, then A(x) and F(x) differ by a constant: A(x) = F(x) + C.

At x = a: A(a) = ∫ₐᵃ f(t) dt = 0, so 0 = F(a) + C, meaning C = -F(a).

At x = b: A(b) = ∫ₐᵇ f(t) dt = F(b) + C = F(b) - F(a).

That's Part 2. The integral equals the antiderivative evaluated at bounds.

The Notation: F(x)|ₐᵇ

The expression F(b) - F(a) appears constantly, so we have shorthand:

F(x)|ₐᵇ = F(b) - F(a)

Read this as "F of x, evaluated from a to b."

Example: ∫₀² x² dx = [x³/3]₀² = (8/3) - (0) = 8/3.

Find the antiderivative, plug in the upper bound, subtract the lower bound. Done.

An Example: Area Under a Parabola

Find ∫₀³ x² dx.

The antiderivative of x² is x³/3 (since d/dx[x³/3] = x²).

Apply Part 2: ∫₀³ x² dx = [x³/3]₀³ = (27/3) - (0/3) = 9 - 0 = 9.

Verify: The Riemann sum approach would require computing lim[n→∞] Σ(3i/n)² · (3/n), which after pages of algebra gives... 9.

The Fundamental Theorem reduced pages to one line.

Why This Changes Everything

Before the Fundamental Theorem, integrals were computed geometrically — case by case, curve by curve. Each new problem required fresh ingenuity.

After the theorem, integration became systematic: find an antiderivative, evaluate at bounds.

This is like the difference between counting individual grains of sand and using a formula for the weight of a pile. One is heroic labor; the other is technology.

The theorem made calculus practical. Suddenly, problems that took Archimedes years could be solved by students in minutes.

The Philosophical Point: Rates and Totals

The theorem encodes a deep relationship between two ways of understanding change:

Differential perspective: How fast is this changing right now? Integral perspective: How much has this accumulated over time?

The theorem says these perspectives are equivalent. Either one determines the other.

Know the rate at every instant → integrate to find the total. Know the total as a function of time → differentiate to find the rate.

Every physical process can be viewed either way. Velocity and position. Force and work. Power and energy. Current and charge. The Fundamental Theorem is why we can move freely between these views.

The Chain Rule Version

What if the upper limit isn't just x, but some function g(x)?

d/dx [∫ₐᵍ⁽ˣ⁾ f(t) dt] = f(g(x)) · g'(x)

This is Part 1 combined with the chain rule.

Example: d/dx [∫₀^(x²) sin(t) dt] = sin(x²) · 2x = 2x sin(x²).

The derivative of the integral equals the integrand evaluated at g(x), times g'(x).

Variable Lower Limit

What if the lower limit varies instead?

d/dx [∫ₓᵇ f(t) dt] = -f(x)

The negative sign appears because increasing x shrinks the integration region.

If both limits vary: d/dx [∫ᵧ₍ₓ₎^h(x) f(t) dt] = f(h(x)) · h'(x) - f(g(x)) · g'(x)

Upper contributes positive; lower contributes negative.

The Existence Question

Part 2 requires finding an antiderivative F. But does every continuous function have one?

Part 1 answers yes. Define:

F(x) = ∫ₐˣ f(t) dt

This is automatically an antiderivative of f, by Part 1.

So every continuous function has an antiderivative. The integral itself provides it.

The catch: this doesn't mean you can write F in terms of elementary functions. The antiderivative of e^(-x²), for instance, exists but can't be expressed using standard functions. You can compute it numerically or define a new function (the error function erf) to name it.

Summary

Part 1: d/dx [∫ₐˣ f(t) dt] = f(x). Differentiating an integral recovers the integrand.

Part 2: ∫ₐᵇ f(x) dx = F(b) - F(a). Integrals can be computed via antiderivatives.

Together: Differentiation and integration are inverse operations.

This is why calculus works. This is why we can solve differential equations, calculate areas and volumes, model physical systems, and transform between rates and totals effortlessly.

One theorem. The entire infrastructure of applied mathematics rests on it.

Further Reading

• Bressoud, D. M. A Radical Approach to Real Analysis. Deep historical context.
• Spivak, M. Calculus. Rigorous proof of the theorem.
• Strogatz, S. Infinite Powers. The history of calculus, accessibly told.

This is Part 2 of the Integrals series. Next: "Indefinite Integrals" — finding antiderivatives, the practical heart of integration.

Part 2 of the Calculus Integrals series.

Previous: What Is an Integral? The Mathematics of Accumulation Next: Indefinite Integrals: Finding Antiderivatives